Chemical-mechanical planarization controller

ABSTRACT

The invention provides a model-based control approach to chemical-mechanical planarization (CMP) control. The preferred embodiment comprises mathematical models of the CMP process. These models play a critical role in obtaining superior control performance. Model-based Control Design involves the construction of a dynamic mathematical model of the system to be controlled, e.g. a removal rate model of a CMP system. The model can then be evaluated via computer simulations, and validated using data from the system. The invention provides a method and apparatus that processes in-situ data from a suite of real-time sensors and produces real-time commands to multiple actuators, such as applied pressures, slurry-flow rate, and wafer/pad velocity. A key aspect of the invention is an integrated model-based pressure-temperature-velocity-slurry flow control system that includes many innovations in real-time mode identification, real-time gain estimation, and real-time control.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. application Ser. No.10/751,228 filed Dec. 30, 2003.

BACKGROUND OF THE INVENTION

1. Technical Field

The invention relates to chemical-mechanical planarization. Moreparticularly, the invention relates to a chemical-mechanicalplanarization controller.

2. Description of the Prior Art

Chemical-Mechanical Planarization (CMP) is an important step in theprocessing of semiconductor wafers and is playing an increasinglycritical role in semiconductor microelectronics fabrication (see TheNational Technology Roadmap for Semiconductors, Semiconductor IndustryAssociation, San Jose, Calif. 1997; J. M. Steigerwald, S. P. Murarka, R.J. Gutman, Chemical Mechanical Planarization of MicroelectronicMaterials, Wiley Interscience, 1997; and W. J. Patrick, W. L. Guthrie,C. L. Stanley, and P. M. Schiable, Application of Chemical MechanicalPolishing to the Fabrication of VLSI Circuit Interconnection, J.Electrochem. Soc., 138, 1778-1784, 1991).

CMP is a process for material removal that uses chemical and mechanicalactions to produce a planar mirror-like wafer surface for subsequentprocessing. For a nominally uniform wafer, CMP is capable of producingan atomically-smooth and damage-free surface at feature level, which isa basic requirement for semiconductor fabrication below 0.25μ (see TheNational Technology Roadmap for Semiconductors, Semiconductor IndustryAssociation, San Jose, Calif. 1997). The superiority of CMP overtraditional etchback techniques with respect to defect reduction andyield enhancement has been demonstrated (for application to tungsten,see K. Wijekoon et al., Tungsten CMP Process Developed, Solid StateTechnology, April 1998). CMP also has fewer processing steps as comparedto traditional etchback methods. CMP is also an enabling technology fortransition to copper interconnects. Optimal CMP maximizes planarity andminimizes oxide erosion and dishing.

Integrated Circuit (IC) makers continue to adopt CMP for advancedmanufacturing, and CMP has now joined standard processing techniques,such as deposition, etch, and lithography in strategic importance.State-of-the-art Application-Specific Integrated Circuits (ASIC) chips,and advanced Dynamic Random Access Memories (DRAMs) are among the latestapplications where CMP is being used. Planarization of features on asemiconductor wafer is a critical factor in Ultra Large-ScaleIntegration (ULSI) processing (0.25μ) for fabrication of multi-levels ofwiring and for trench isolation. As device geometries shrink, there areincreasingly more stringent requirements on deposition, etch, andlithography due to increases in aspect ratio of device structures. Thereis a lithography constraint on the step height, i.e. feature variationsthat require the pattern entirely be confined to within a depth of focusof ±0.3μ. For DRAM applications, planarization processes for trenchisolation require thickness to be controlled within ±0.1μ or better.This requirement when achieved over all features is referred to asglobal planarization. For integrating CMOS technologies of a quartermicron (0.25μ or below), CMP is being used in advanced applications suchas Shallow Trench Isolation (STI).

Description of the CMP Process

One distinguishes different kinds of CMP systems by its kinematicmotions, e.g. rotational, orbital or linear CMP systems. A schematic ofa typical rotational CMP machine is shown in FIG. 1 (see, e.g. J. M.Steigerwald, S. P. Murarka, R. J. Gutmann, Chemical MechanicalPlanarization of Microelectronic Materials, Wiley Interscience, 1997).The rotating wafer 10 borne by a wafer carrier 11 rests on a rotatingpad system 12, consisting of one or more pads. The pad system is part ofa polishing table 13. A pressurized retaining ring surrounds the waferand holds it in place. A nominally uniform load pressure distributionacts on the wafer. For oxide or silicon polishing, an alkaline slurry 14of colloidal silica is continuously fed to the wafer/pad interface.Although the detailed mechanisms are under investigation, a surfacelayer forms as a result of chemical processes, and the resultingreaction product is removed by the mechanical abrasive action of the padand the slurry. The behavior is most complex at the edge of the wafer.The differential velocity and pressures, as well as slurry composition,determine the local removal rate. The dynamic nature of the deformationof the pad determines the local pressure gradients across the wafer andthe resulting planarization uniformity. To planarize features across thewhole wafer evenly, the material removal rate across the wafer must beuniform.

State-of-the-Art CMP Process Control

The goal of CMP processing is to achieve a specified thickness anduniformity in a repeatable fashion. Major problems in CMP includecontrolling the material removal or, equivalently, the material removalrate, and the uniformity on each run, and reproducibility fromrun-to-run. Typically an in-situ sensor is used to detect the end-pointof the process, i.e. to detect when the desired amount of material isremoved, at which point in time the process is stopped.

A widely used approach for controlling CMP performance involves thefollowing two-step trial-and-error process (see, e.g., R. Allen, C.Chen, K. Lehman, R. Shinagawa, V. Bhaskaran, CMP: Where Does It End,Yield Management Solutions Magazine, Vol. 4, No. 1, 2002):

-   -   (1) process parameters are adjusted to give good uniformity, and    -   (2) end-point control using an in-situ rate sensor is used to        achieve desired material removal thickness.

From a control theory perspective, this approach is called Open-loopcontrol, because the control variables are not adjusted during the run.Neither are these control variables tuned from run to run, but heldconstant. This approach has at least the following limitations:

First, the process operating window is very narrow, because the processis finely tuned to generate a recipe where the input process parametervalues yield acceptable uniformity for most materials. Therefore, theprocess performance is not robust, being very sensitive to disturbancesand input variations, such as pad wear, temperature variations, slurryconcentration, sensor drift, etc. Furthermore this approach does notwork well for different materials.

Second, if the output specifications for the planarization are changed,then considerable trial-and-error is required to re-establish the inputoperating conditions necessary to obtain uniformity. These limitationsrequire intensive process monitoring as well as the availability of many(expensive) test wafers.

One approach that addresses some of these limitations is calledRun-to-run control. In run-to-run control, the control variables areheld constant during the run, but may be modified between runs based onin-line and/or ex-situ (post-process) measurements. This approach workswell for compensating slow drifts such as pad wear or slow temperaturevariations, but does not work for wafer-to-wafer variations such asvariations in incoming thickness profile, variations in slurryconcentrations, etc.

It would be advantageous to provide an approach that addresses all ofthese limitations.

SUMMARY OF THE INVENTION

The presently preferred embodiment of the invention, which provides anapproach that addresses all of these limitations, is referred to hereinas a Model-Based Feedback/Feedforward control approach. This type ofcontrol is called Feedback/Feedforward because the control variables areadjusted during the run based on in-situ measurements. This type ofcontrol is unprecedented in the history of CMP applications. For thispurpose, the dynamic behavior of the CMP system during a run has to betaken into account. The inventors have developed a detailedphysics-based dynamic mathematical model of the CMP process (includingchemistry), as well as reduced dynamic models for control. These modelsplay a critical role in obtaining superior control performance.Model-based control design involves the construction of a physics-basedmathematical model of the system to be controlled, e.g. a removal ratemodel of a CMP system. The model can then be evaluated via computersimulations and validated using experimental data from the system.

The order and complexity of the model depends on the application.Typically, the order is large for equipment design evaluation purposes.A reduced-order model is constructed for feedback control system design.The closed-loop control system is first evaluated via a computersimulation. Once satisfactory results are obtained with the simulation,the feedback controller is used to control the actual system. Thisapproach not only provides physical insight into the open-loop andclosed-loop behavior of the system, but also can be used to extractmaximum performance from a given system. Because a physics-basedmathematical model is constructed, modifications of the system can beevaluated via computer simulations prior to any hardware modifications.In essence, one can construct a virtual engineering environment, whichcan be used to evaluate and optimize system designs before expensiveequipment or hardware is purchased.

The use of custom embedded feedback control is becoming more critical insemiconductor manufacturing equipment. The herein disclosed model-basedfeedback/feedforward control design technology provides a systematicprocess for modeling, simulation, and controller design. The approachherein disclosed has the capability to extract maximum performance fromcomplex multi-input multi-output processes, which have a high degree ofinteraction between various process inputs and outputs. Traditionalsingle-input single-output design approaches would limit the kind ofperformance that can be achieved in systems with strong coupling betweenvarious input and/or output variables.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic of a typical rotational CMP machine;

FIG. 2 is a block schematic diagram of an advanced dynamic CMP modelaccording to the invention;

FIG. 3 is an example of a multi-zone pressure actuation model accordingto the invention;

FIG. 4 is a block schematic diagram of an exemplary CMP controlleraccording to the invention;

FIG. 5 is a block schematic diagram showing a temperature control moduleaccording to the invention;

FIG. 6 is a graph showing typical normalized base vectors according tothe invention;

FIG. 7 is a typical plot showing the resulting mesh as a function of c₂and c₄ for Equation (5) according to the invention; and

FIG. 8 is a block schematic diagram showing a pressure profile controlmodule according to the invention.

DETAILED DESCRIPTION OF THE INVENTION

In-Situ Sensing (25, 27)

A recent development in CMP is the use of real-time in-situ sensors,such as optical or optical-eddy-current sensors 25 to monitorwafer-scale as well as die-scale uniformity, and allow real-timefeedback control of wafer uniformity. The primary objective is tocontrol global planarity. Therefore, it is necessary to sense variations(non-uniformity) in the removal rate at the wafer-scale as well asdie-scale. Off-line metrology can be used to monitor both wafer as wellas die-scale uniformity, and the resulting measurements can be used forrun-to-run control. Several sensors have been proposed for and used inCMP for monitoring the material removal rate. See M. Sun, H.-M. Tzeng,H. Litvak, D. Glenn, In-situ Detection of Film Thickness Removal DuringCMP of Oxide and Metal Layers, Proc. CMP-MIC, February 1996;S. Inaba,et. al., Study of CMP Polishing Pad Control Method, CMP-MIC Conference,1998 IMIC-300P/98/0044, pp. 44-51, 1998; KLA-Tencor, Press Release, Mar.5, 2001; G. Dishon, et. al., On-Line Integrated Metrology for CMPProcessing, VMIC Specialty Conferences, CMP Planarization, pp. 1-5,1996; L. Chen, C. Diao, A Novel In-Situ Thickness Measurement MethodUsing Pad Temperature Monitoring For CMP Technology, CMP-MIC Conference,1996 ISMIC-100P/96/0241, pp. 241-248, 1996; M. Sun, H.-M. Tzeng, H.Litvak, D. Glenn, In-situ Detection of Film Thickness Removal During CMPof Oxide and Metal Layers, Proc. CMP-MIC, February 1996; S. Inaba, et.al., Study of CMP Polishing Pad Control Method, CMP-MIC Conference, 1998IMIC-300P/98/0044, pp. 44-51, 1998.

It is anticipated that the invention herein disclosed may as well beapplied to augment some of the existing optical sensors with patternrecognition capabilities that enable these sensors to monitorwafer-scale as well as die-scale uniformity in real-time. Othersophisticated sensor technologies will most certainly be introduced inthe future for CMP closed-loop in-situ control. The invention herein islikewise applicable with these new sensors.

Advanced CMP Model (21)

FIG. 2 is a block schematic diagram of an advanced dynamic CMP model 21according to the invention. The physical processes involved in CMP are acomplex combination of chemistry, abrasion, and mechanics. At thehighest level, a global model has externally controllable variables,e.g. applied pressures, slurry flow rate, wafer/platen velocity, etc.,designated as inputs and predicts the material removal rate RR(x, t) asa function of time t, and location on the wafer x, as well as thetemperature T(t) as a function of time. This global model is formed froma combination of several different component models. Specifically, acontact mechanical model 20 predicts the contact pressure and relativevelocity between the pad and the wafer surface. A chemical andmechanical model 22 predicts the local removal rate as a function oflocal slurry, pad, and wafer properties. A transport model 24 predictsthe distribution of slurry, pad, and wafer properties across thewafer/pad interface. Finally, a thermal model 26 predicts thetemperature distribution at the wafer/pad interface necessary foraccurate chemistry and transport modeling. For purposes of controllingWithin-Wafer Non-Uniformity (WIWNU), the wafer-scale models are ofgreatest interest and not die- or feature-scale models. These fourmodels are interdependent because local polishing and transport dependon both pressures and velocities; temperature depends on friction andchemistry; and chemistry and transport depend on temperature. Thus, thedivision between models is primarily based on the different physicalprocesses each represents.

The mechanical and kinematic model for the contact pressure and relativevelocity between the wafer and pad is fairly well understood (see G. Fu,A. Chandra, S. Guha, G. Subhash, A Plasticity-Based Model of MaterialRemoval in Chemical-Mechanical Polishing (CMP), IEEE Transactions onSemiconductor Manufacturing, Vol. 14, No. 4, 406-417, November 2001; andG. Fu, A. Chandra, Effects of Viscoelastic Pad Deformation on MaterialRemoval Rate in Chemical Mechanical Planarization, Proc. CMP-MICConference, pp. 67-76, February 2002). Some primary uncertainties arethe mechanical properties of the pads and the effective friction betweenthe pad and wafer. Edge effects have been successfully modeled, but dueto their high sensitivity to pad properties, it is common to augment ortune pure physics-based mathematical models with experimental results tomatch edge effects. This approach of starting with a basic model thataccurately represents the main physics of the problem and tuning withexperimental results has produced excellent semiempirical models forcontrol design. The mechanical model supports generic multi-zonepressure actuation. An example of a multi-zone pressure actuation modelis shown in FIG. 3.

The polishing model involves predicting the removal rate at a given timeand wafer location as a function of contact pressure, relative wafer/padvelocity, abrasive particle concentration and size distribution, andslurry/wafer chemistry. The mechanical, thermal, and transport modelsare used to determine the value of the variables at all times andlocations on the wafer. In CMP, as compared to pure mechanicalpolishing, the slurry reacts with the wafer to form a thin layer on thewafer surface with modified composition and morphology (see G. Fu, A.Chandra, S. Guha, G. Subhash, A Plasticity-Based Model of MaterialRemoval in Chemical-Mechanical Polishing (CMP), IEEE Transactions onSemiconductor Manufacturing, Vol. 14, No. 4, 406-417, November 2001; J.Luo, D. A. Dornfeld, Material Removal Mechanism in Chemical MechanicalPolishing: Theory and Modeling, IEEE Transactions on SemiconductorManufacturing, Vol. 14, No. 2, pp. 112-133, May 2001; and J. Luo, S.Aksu, D. A. Dornfeld, Material Removal Regions in Chemical MechanicalPolishing: Coupling Effects of Slurry Chemicals, Abrasive SizeDistributions and Wafer-Pad Contact Area, Proc. CMP-MIC Conference, pp.49-58, February 2002). This thin layer is then mechanically removed byabrasion by the particles in the slurry.

In general, there are two limiting modes, a hydrodynamical contact modeand a solid-solid polishing mode. If the pressure is low and therelative velocity is high, the wafer does not touch the pad, but insteadrides on a thin layer of slurry. Abrasion occurs when particles withinthe slurry impact the wafer. In the solid-solid polishing mode the wafercontacts the pad and abrasion occurs by particles partially embedded inthe pad. Models for the solid-solid contact mode are fairlywell-developed (see J. Luo, D. A. Dornfeld, Material Removal Mechanismin Chemical Mechanical Polishing: Theory and Modeling, IEEE Transactionson Semiconductor Manufacturing, Vol. 14, No. 2, pp. 112-133, May 2001;and J. Luo, S. Aksu, D. A. Dornfeld, Material Removal Regions inChemical Mechanical Polishing: Coupling Effects of Slurry Chemicals,Abrasive Size Distributions and; Wafer-Pad Contact Area, Proc. CMP-MICConference, pp. 49-58, February 2002), but there is still considerableresearch being done.

The transport model describes the distribution of slurry and abrasiveparticles across the pad. Factors such as wafer rotation rate, appliedpressures, and location and slurry flow rate all affect the transport ofthe slurry. Because the slurry viscosity is somewhattemperature-dependent, transport may have a small dependence ontemperature.

As material is removed from the wafer the chemical bonds are broken,thus releasing energy. Similarly, viscous dissipation in the slurryreleases energy that affects the local temperature of the slurry, pad,and wafer. Therefore, the temperature of the slurry, pad, and wafer allchange during the polishing process. Because the chemistry between theslurry and the wafer is usually fairly strongly dependent ontemperature, a temperature model improves the polishing model.

Exemplary Model-Based CMP Process Controller

An exemplary CMP controller 40 consists of five modules, see FIG. 4.These modules are described in the following discussion.

Temperature Control Module (41)

Many slurries for copper CMP are temperature sensitive, i.e. the copperremoval rate depends on the local temperature of the wafer and pad, seeH. Chiou, Z. Lin, L. Kuo, S. Shih, L. Chen C. Hsia, Thermal Impact andProcess Diagnosis of Copper Chemical Mechanical Polish, Proc. IEEEInternational Interconnect Technology Conference, pp. 83-85, 1999; andD. White, J. Melvin D. Boning, Characterization and Modeling of DynamicThermal Behavior in CMP, J. The Electrochemical Society, Vol. 150, No.4, pp. 271-278, 2003. The temperature varies across the wafer as coolslurry flows in one side and is heated by frictional heat generationbetween the wafer and pad. If one could measure wafer temperature atseveral points on the wafer, one would see smoothly varying mode-shapesas a function of time and actuation variables that influencetemperature, such as applied pressures, slurry flow rate and motorvelocities. These smoothly varying time- and actuation-dependenttemperature mode-shapes determine in part the local removal rate on thewafer.

The herein disclosed temperature control module controls the average padtemperature, for example responsive to temperatures measured by anin-situ temperature sensor 27 and a thermal model (discussed above; seeFIG. 2), such that the resulting wafer profile is as uniform asphysically possible at the end of the polish. See FIG. 5, which shows aschematic of the temperature control module.

Mode Shape Identification (51)

First, one has to define what it means to obtain a wafer profile that isas uniform as physically possible, i.e. maximum planarization achievableby the equipment. There are several metrics to measure the uniformity ofa wafer. A well-known metric is Within-Wafer-Non-Uniformity (WIWNU),which basically is defined as the I₂-norm (Euclidean vector norm) of theprofile divided by its average. Another metric is defined as thedifference between the maximum and the minimum thickness across thewafer. The data points at the edge of the wafer are usually excludedfrom these calculations, e.g. a 3-5 mm edge exclusion zone. Thepresently preferred metric for wafer uniformity is the I₂-norm of a2^(nd) and 4^(th) order identified mode shape of the wafer profile. Thisis explained below.

It is assumed that a new diameter scan of wafer thickness is availableeach time the controller is called. Each time a new measurement isavailable, the measured wafer profile is approximated by fitting an nthorder polynomial through this data using standard least-squares. Leth(r) denote the measured data as a function of position r across adiameter scan. Let p(r) denote an n^(th) order polynomial in r:$\begin{matrix}{{{p(r)} = {\sum\limits_{k = 0}^{n}{c_{k}{\overset{\_}{r}}_{k}}}},{{where}\text{:}}} & (1) \\{\quad{{{{\overset{\_}{r}}_{k} = \frac{r_{i}^{k}}{\quad{\overset{\_}{m}}_{k}}},{i = {- N}},{\ldots\quad 0},\ldots\quad,{N;{k = 0}},\ldots\quad,{n;}}{{\overset{\_}{m}}_{k} = {\max\limits_{i}{\left( r_{i}^{k} \right).}}}}} & (2)\end{matrix}$

The vectors {overscore (r)}_(k), k=0 . . . n, are called normalized basevectors or mode shapes of the profile p(r). The coefficients c_(k), k=0. . . n are called mode shape coefficients. In the following n=4 is usedsince that typically provides enough resolution to model a waferprofile. However, the inventive method is not limited to n=4. With n=4,Equation (1) becomes:p(r)=c ₀ {overscore (r)} ₀ +c ₁ {overscore (r)} ₁ +c ₂ {overscore (r)} ₂+c ₃ {overscore (r)} ₃ +c ₄ {overscore (r)} ₄.   (3)

FIG. 6 shows the normalized base vectors for this case. The mode shapecoefficients c_(k), k=0 . . . 4 are estimated by solving:$\begin{matrix}{{\underset{\underset{\quad\overset{\_}{h}}{︸}}{\begin{bmatrix}{h\left( r_{- N} \right)} \\\vdots \\{h\left( r_{N} \right)}\end{bmatrix}} - {\underset{\underset{R}{︸}}{\begin{bmatrix}1 & {{\overset{\_}{r}}_{1}\left( {- N} \right)} & {{\overset{\_}{r}}_{2}\left( {- N} \right)} & {{\overset{\_}{r}}_{3}\left( {- N} \right)} & {{\overset{\_}{r}}_{4}\left( {- N} \right)} \\\vdots & \vdots & \vdots & \vdots & \vdots \\1 & \overset{\quad}{\quad{{\overset{\_}{r}}_{1}(N)}} & {\quad{{\overset{\_}{r}}_{2}(N)}} & {\quad{{\overset{\_}{r}}_{3}(N)}} & {\quad{{\overset{\_}{r}}_{4}(N)}}\end{bmatrix}} \cdot \underset{\underset{\quad\overset{\_}{c}}{︸}}{\begin{bmatrix}c_{0} \\c_{1} \\c_{2} \\c_{3} \\c_{4}\end{bmatrix}}}} = {\begin{bmatrix}0 \\\vdots \\0\end{bmatrix}.}} & (4)\end{matrix}$

By solving this standard least-squares problem at each time-step whennew measurements are available and storing the results, one can monitorthe evolution of the mode shape coefficients as a function of time, t:c₀(t), c₁(t), etc. The first coefficient, c₀, is an estimate of theaverage or mean of the measured profile h(r). The coefficients c₁ and c₃are estimates of the asymmetry in the profile h. Because the wafer isrotating, it is assumed that any asymmetry on the wafer is eventuallyaveraged out to zero. The coefficients c₂ and c₄ determine theuniformity of the profile h, and are therefore the coefficients ofinterest. A large negative coefficient c₂ typically denotes an edge-thinprofile and a large positive coefficient c₂ typically denotes anedge-thick profile. Likewise, the coefficient c₄ contains informationabout the steepness of the edge. If c₄ is relatively large with respectto c₂, typically the profile h has a uniform center with steep edge. Ifc₄ is relatively small with respect to c₂, typically the profile hgradually increases or decreases from center to edge. A good metric ofthe uniformity of the profile h is the I₂-norm of the approximateprofile determined by the coefficients c₂ and c₄: $\begin{matrix}\begin{matrix}{{{{c_{2}{\overset{\_}{r}}_{2}} + {c_{4}{\overset{\_}{r}}_{4}}}}_{2} = \sqrt{\sum\limits_{i = {- N}}^{N}\left( {{c_{2}\frac{r_{i}^{2}}{\quad{\overset{\_}{m}}_{2}}} + {c_{4}\frac{r_{i}^{4}}{\quad{\overset{\_}{m}}_{4}}}} \right)^{2}}} \\{{= \sqrt{\sum\limits_{i = {- N}}^{N}\left( {{c_{2}^{2}\frac{r_{i}^{4}}{\quad{\overset{\_}{m}}_{2}^{2}}} + {2c_{2}c_{4}\frac{r_{i}^{6}}{\quad{{\overset{\_}{m}}_{2}{\overset{\_}{m}}_{4}}}} + {c_{4}^{2}\frac{r_{i}^{8}}{\quad{\overset{\_}{m}}_{4}^{2}}}} \right)}}\quad} \\{= {\sqrt{{c_{2}^{2}\underset{\overset{\quad}{\quad}}{\underset{\underset{v_{\alpha}}{︸}}{\frac{1}{\quad{\overset{\_}{m}}_{2}^{2}}{\sum\limits_{i = {- N}}^{N}r_{i}^{4}}}}} + {c_{2}c_{4}\underset{\underset{v_{\beta}}{︸}}{\frac{2}{\quad{{\overset{\_}{m}}_{2}{\overset{\_}{m}}_{4}}}{\sum\limits_{i = {- N}}^{N}r_{i}^{6}}}} + {c_{4}^{2}\underset{\underset{v_{\gamma}}{︸}}{\frac{1}{\quad{\overset{\_}{m}}_{4}^{2}}{\sum\limits_{i = {- N}}^{N}r_{i}^{8}}}}}.}}\end{matrix} & (5)\end{matrix}$

The scalars v_(α), v_(β) and v_(γ) can be computed a priori and storedin memory. Independent of the incoming profile h, one can computeEquation (5) for a given range of c₂ and c₄ values and plot theresulting mesh as a function of c₂ and c₄, see FIG. 7. The isonorms inthis figure form rotated ellipses because of the quadratic form ofEquation (5). The mesh in FIG. 7 has a global minimum at the point (c₂,c₄)=(0, 0). The mesh has an infinite number of local minima along thedotted line that is defined by the angle of rotation of the ellipse,i.e. for any given value c₂, there is a unique value c₄ that yields theminimum I₂-norm of the profile determined by c₂ and c₄. This line ofminima can be parameterized as: $\begin{matrix}{{{c_{4} = {\alpha \cdot c_{2}}},{with}}{\alpha = {{- \tan}\quad{\left( {\frac{\pi}{2} - \left( {\frac{\pi}{4} - {\frac{1}{2}{\tan^{- 1}\left( \frac{v_{\alpha} - v_{\gamma}}{v_{\beta}} \right)}}} \right)} \right).}}}} & (6)\end{matrix}$

For N=100 in Equation (5), the value of a in Equation (6) is −1.345. ForN=150, the value of α is −1.349, i.e. α is not a strong function of thevalue of N.

It is now possible to give a mathematical answer to the question “Whatdoes as uniform as physically possible mean?” A profile with c₂ and c₄both equal to zero is as uniform as theoretically possible, i.e. astraight perfectly horizontal profile, but this unique minimum is notalways physically possible because there are a limited number ofactuators. It is likely that either c₂ can be zero or c₄ can be zero,but it is unlikely that both are zero at the same time. However, as canbe seen from FIG. 7, if only one of the two is zero, the minimum norm ofthe final resulting profile is not achieved. Therefore, if one of thecoefficients is non-zero, the minimum norm of the final resultingprofile is achieved only if the other coefficient is such that the point(c₂, c₄) is on the line given by Equation (6). If that can beaccomplished, the physically best-possible profile uniformity isachieved. Driving the profile toward a point on this line is the task ofthe Master Control Loop 53, see FIG. 5.

Master Control Loop (53)

To drive the wafer profile to a state such that the point (c₂, c₄) is onthe line given by Equation (6), define the time-dependent controlfunction s(t):s(t)=c₄(t)−α·c ₂(t),   (7)with α as defined in Equation (6). The objective of driving (c₂, c₄)toward the line defined by Equation (6) can now be replaced by theobjective of driving the control function s(t) toward zero over time(and keep it at zero). To drive the control function to zero in a finiteamount of time, the rate of s(t) has to be such that: $\begin{matrix}{{\frac{\mathbb{d}{s(t)}}{\mathbb{d}t} \leq {{- \eta}\quad{for}\quad{s(t)}} > 0},{\frac{\mathbb{d}{s(t)}}{\mathbb{d}t} \geq {{+ \eta}\quad{for}\quad{s(t)}} < 0.}} & (8)\end{matrix}$for some positive constant η. This condition can be reformulated as:$\begin{matrix}{{{{s(t)}\frac{\mathbb{d}{s(t)}}{\mathbb{d}t}} \leq {{- \eta}{{s(t)}}}},{\eta > 0.}} & (9)\end{matrix}$

If the condition of Equation (9) is met, the control of s(t) is aconverging control.

Again, the invention controls (or influences) the profile s(t) bycontrolling average wafer/pad temperature. Temperature, in turn, cannotbe controlled directly, but is controlled by adjusting the inputpressures, slurry flow rate, and wafer/pad velocities, see FIG. 2. Inother words, if the master control loop requests a certain temperature,the slave loops translate that to corresponding values of pressure,slurry flow rate, and velocities. For this reason, it is critical tohave a sufficiently accurate dynamic model that relates pressure, slurryflow rate, and velocities to wafer/pad temperature.

Dynamic Temperature Model (52)

The two blocks in FIG. 5, Dynamic Temperature Model 52 and DynamicTemperature Reference Model 54 both represent a dynamic model thatdescribes the evolution of average wafer/pad temperature over time. TheDynamic Temperature Model describes the average wafer/pad temperatureand can be replaced by a temperature measurement if a temperature sensor27 is available, or used in an estimator to improve noisy measurements.The Dynamic Temperature Reference Model describes desired or referencewafer/pad temperature and is driven by the master control loop.

The dynamic temperature model is an energy balance for a volume at thewafer/pad interface. This balance says that the rate of change ofinternal energy in that volume is equal to the net power transportedinto the volume. The net power into the volume includes three terms: 1)conduction to and from the slurry and surroundings, 2) slurry transportinto and out of the volume, and 3) power generation through friction atthe wafer/pad interface. The first term is proportional to the averagetemperature difference between the interface and the slurry orsurroundings. The second term is proportional to slurry flow. The thirdterm is proportional to the applied pressure. The latter two termsrelate actuator inputs (slurry flow, velocity and pressure) to the padtemperature. Uncertain parameters in this model can be adjusted usingexperiments to provide agreement between the model and the system. Theresulting expression is of the form $\begin{matrix}{\frac{\mathbb{d}T}{\mathbb{d}t} = {{K_{c}\left( {T - T_{s}} \right)} + {K_{s}Q_{s}} + {K_{f}{Q_{f}.}}}} & (10)\end{matrix}$

Here T and T_(s) are the interface and slurry or surroundingtemperatures, respectively. The slurry flow rate into the interfacebetween the pad and wafer depends on the specific transport anddistribution of slurry, and is described by the quantity Q_(s). Thefrictional power dissipation between the slurry and pad is Q_(f) and isgenerally a function of the rotation rate and applied pressure on thepad. The three empirical constants (K_(c), K_(s), K_(f)) are determinedexperimentally for a particular application.

The frictional power generation, Q_(f) can be measured directly bymeasuring the electrical power, e.g. motor current, used to drive therotation motors. After correcting for dissipation in the motors andbearings, and for acceleration terms during unsteady operation, whatremains is the frictional power at the wafer/pad interface. Thisobservation is important and novel. It indicates that the thermal modeldescribed above can also be the basis for a temperature sensor since itdirectly relates temperature to the frictional power dissipation througha relatively simple ordinary differential equation, Equation (10).

Dynamic Temperature Reference Model (54)

As mentioned above, the Dynamic Temperature Reference Model 54 describesdesired or reference wafer/pad temperature and is driven by the mastercontrol loop. In effect, it runs the same temperature model asdescribed, except that it is not driven by the pressures p(t)—apart fromthe initialization stage, but by a desired motor current set pointI_(spt)(t). After the initialization stage, the reference temperatureT_(ref)(t) calculated from the reference temperature model representsthe desired CMP process temperature at every time instant. Based on theresults from the mode shape identification, the master control loopdetermines the desired temperature and decides when and how to get atthat temperature by changing the motor current set point I_(spt)(t)accordingly.

Pressure Slave Loop (57)

The pressure slave loop 57 is a feedback controller combined withfeedforward control based on the Dynamic Temperature Model 52, see, e.g.G. F. Franklin, J. D. Powel, A. Emami-Naeini, Feedback Control ofDynamic Systems, 4^(th) edition, Prentice Hall, 2002. The inputs to thepressure slave loop are the process temperature, T(t), either from asensor or from the model 52, the desired reference temperature,T_(ref)(t), and the motor current set point I_(spt)(t), respectively,see FIG. 5. First, a nominal (feedforward) set point pressure iscomputed by substituting the desired motor current set point into themodel 52 and back calculating the corresponding desired pressures. Next,a pressure correction is added by feeding the tracking errore(t)≡T_(ref)(t)−T(t) into the feedback controller. The resulting outputpressures are called offset pressures, p_(offs)(t), which feed into thePressure Profile Control Module 42, see FIG. 4.

Slurry Flow and Velocity Slave Loop (55)

Similar to the pressure slave loop, the slurry flow and velocity slaveloops are fed by set point values coming out of the master control loop.Slurry flow and velocity are directly influenced by temperatureaccording to the Dynamic Temperature Model 52. They can therefore beused to control temperature as well. These two controls have to be usedwith great care though. The main use of slurry flow is not to controltemperature, but to add the right chemistry composition to the CMPprocess, see FIG. 2. Control of temperature is a secondary function ofslurry flow. However, the master control loop can determine that thepressure slave loop is not able to control the profile adequately, inwhich case it can decide to activate the slurry flow slave loop to helpthe pressure slave loop. For example, if a process runs too hot for toolong, the pressure slave loop likely saturates the pressures low, whichslows down the polish rate according to Preston's equation, see F. W.Preston, The Theory and Design of Plate Glass Polishing Machines, J.Soc. Glass Technology, Vol. 11, pp. 214-256, 1927. If the pressuressaturate low for too long and the temperature is still too high, it isadvantageous to increase the slurry flow to remove heat and cool theprocess to its desired temperature.

Similarly, increasing or decreasing the velocity increases or decreasesthe process temperature. Velocity, however, also affects the polish rateaccording to Preston's equation, and should therefore be handled withcare. Furthermore, velocity typically also determines the rate of dataacquisition of the sensors. Changing the velocity throughout the polishchanges this acquisition rate, and care has to be taken to handle thischange correctly.

Similar to the outputs of the pressure slave loop, the outputs of theslurry flow and velocity slave loop, are offset slurry flow and offsetvelocity feeding into the Slurry Flow and Motor Velocity Control Module43, respectively, see FIG. 4.

Pressure Profile Control Module (42)

The Pressure Profile Control Module 42 controls the individual zonepressures for a multi-zone pressure CMP process, see FIG. 3 and FIG. 8.The herein disclosed Pressure Profile Control Module is the first of itskind to perform multivariable in-situ pressure feedback usingmultivariable in-situ wafer thickness measurements, as obtained with anin-situ thickness sensor 25 (see FIG. 2). One goal is to adjust thepressures in-situ such that a uniform wafer thickness is obtained at theend of each polish, independent of incoming wafer profile and/or processdisturbances.

Zone-Averaging Model (83)

One goal of the zone-averaging model 83 is to extractmore-or-less-independent thickness information from the measured waferprofile h(r), r=31 N . . . N, for each of the pressure zones. Assumethat there are n independent pressure zones. Although not required, itis desired to extract n different thickness measurements. The mainreason for extracting the same number of measurements as there arepressure zones is that this way the feedback controller becomes aMulti-Input-Multi-Output (MIMO) controller with equal number of inputsand outputs (this is called a square MIMO controller), which makes thecontrol design more convenient. If there are more pressure actuationzones than thickness measurements, the pressure control is typically notunique because there is an infinite number of pressure combinations thataffect the thickness measurement in the same way. If there are morethickness measurements than pressure actuation zones, the pressurecontrol typically is not able to achieve an independent objective forall thickness measurements. An equal number of thickness measurements(sensors) and pressure zones (actuators) is therefore typically the bestchoice.

It is not obvious how to extract n measurements from 2N data pointswhere n<<2N. One way is to divide the profile into n sections thatcorrespond more-or-less to the physical locations of the pressure zonesand then take an unweighted average in each zone. This is a good firstattempt but typically does not result in the most uniform profile asmeasured by, e.g. the Within-Wafer-Non-Uniformity (WIWNU) metric. Theproblem is that it is desirable that the entire profile of 2N points beas uniform as possible, but one can only control n independent zones. Ifa 300 mm (12-inch diameter) wafer is measured with measurements spread 2mm apart, then N=75. A multi-zone pressure head could have as few as twoand perhaps at most ten independent pressure zones, which makes it ahard problem to extract two to ten measurements out of 150 data pointsfor pressure feedback such that all 150 data points are as uniform aspossible at the end of the polish. The herein disclosed Pressure ProfileControl Module implements an innovative model-based way for selecting nmeasurements from 2N data points.

The mechanical/kinematic model shown in FIG. 2 models the contactpressure between wafer and pad in a finite number of nodes as a functionof the input pressures in the different zones. If the number andlocation of these contact nodes selected is equal to the nodes wherewafer thickness is measured, one can compute a gain matrix B thatrelates contact pressure to input pressure in steady-state:$\begin{matrix}{{\begin{bmatrix}p_{N}^{c} \\p_{N - 1}^{c} \\\vdots \\p_{{- N} + 1}^{c} \\p_{- N}^{c}\end{bmatrix} = {\underset{\underset{B}{︸}}{\begin{bmatrix}b_{N\quad 1} & \cdots & b_{Nn} \\b_{N - 11} & \cdots & b_{N - {1n}} \\\vdots & ⋰ & \vdots \\b_{{- N} + 11} & \cdots & b_{{- N} + {1n}} \\b_{{- N}\quad 1} & \cdots & b_{- {Nn}}\end{bmatrix}} \cdot \begin{bmatrix}p_{1} \\\vdots \\p_{n}\end{bmatrix}}},} & (11)\end{matrix}$where p^(c) _(i) denotes the contact pressure in node i, and p_(j)denotes the input pressure in zone j. The contact pressure, in turn, isdirectly related to the removal rate of the CMP process, according toPreston's equation, see F. W. Preston, The Theory and Design of PlateGlass Polishing Machines, J. Soc. Glass Technology, Vol. 11, pp.214-256, 1927: $\begin{matrix}{{\frac{\mathbb{d}{h_{i}(t)}}{\mathbb{d}t} = {{- K_{p}} \cdot p_{i}^{c} \cdot v_{i}^{r}}},{i = {{- N}\quad\ldots\quad N}},} & (12)\end{matrix}$where v^(r) _(i) is the relative velocity between wafer and pad at nodei, and K_(p) is called Preston's constant. One approach in the hereindisclosed invention is to use the columns of the gain matrix B asweight-factors for calculating a weighted average in each zone:$\begin{matrix}{{h_{j}^{avg} = {\left\lbrack {b_{Nj}\quad\ldots\quad b_{- {Nj}}} \right\rbrack \cdot \begin{bmatrix}{h(N)} \\\vdots \\{h\left( {- N} \right)}\end{bmatrix}}},{j = {1\quad\cdots\quad{n.}}}} & (13)\end{matrix}$

One physical interpretation of this way of averaging is that for eachzone the measurements that can be most influenced by the pressureactuation are the measurements that receive the strongest weight in theaverage.

Tracking Error Calculation and Reference Generator (85)

The raw tracking error in each zone is defined as the difference betweena reference value coming out of the reference generator 84, and theaverage as defined by Equation (13):e _(j) ^(raw) =h _(j) ^(ref) −h _(j) ^(avg) , j=1 . . . n.   (14)

There are two modes of control:

-   -   1. Absolute thickness control. In this mode, the reference value        is an external signal specifying a desired thickness and        thickness removal rate, at each point in time. The goal is to        make the absolute thickness and polish rate in each zone equal        to the corresponding external reference values.    -   2. Uniformity control. In this mode, the reference value is        equal to the measured average thickness in one of the zones, at        each point in time. The goal is to control uniformity of the        wafer by making the thickness in the other zones equal to the        thickness in the reference zone.

After passing through a deadband and a limiter, the tracking error ispassed on to the MIMO feedback loop 86.

Multi-Input Multi-Output (MIMO) Feedback Loop (86)

The MIMO Feedback Loop is a truly multivariable controller because themulti-zone pressure system is truly multivariable with multiple pressureactuators, recall FIG. 3, and multiple wafer thickness sensormeasurements, recall Equation (13). Because most multivariable feedbackcontrol design techniques are based on linear models, linear models mustbe derived from the dynamic, possibly non-linear, CMP sub-models (seeFIG. 2) that describe the linear CMP behavior at a specific operatingpoint (a selection of constant input values). The continuous linearmodels take on the form: $\begin{matrix}\begin{matrix}{{\frac{\mathbb{d}{x(t)}}{\mathbb{d}t} = {{{Ax}(t)} + {{Bu}(t)}}},} \\{{{y(t)} = {{{Cx}(t)} + {{Du}(t)}}},}\end{matrix} & (15)\end{matrix}$where x(t) denotes the vector of state variables, u(t) denotes thevector of input variables (actuators), and y(t) denotes the vector ofoutput variables (sensors), and A, B, C, and D are the state-spacematrices of appropriate size. In discrete time, Equation (15) translatesto:x(k+1)=Ax(k)+Bu(k),   (16)y(k)=Cx(k)+Du(k),where k denotes the current (discrete) time sample t_(k)=kΔT, whilet_(k) denotes current time at sample k and ΔT denotes the sampling timeof the model of Equation (16), i.e. the time-interval of discretization.For the CMP pressure model, Equations (15) and (16) are formed byEquations (11-13). The thickness at each node on the wafer, h,(k), i=−N. . . N, form the state vector x(k), the pressures p_(j), j=1 . . . n,form the input vector u(k), and the average measured thickness in eachzone, h_(j) ^(avg), j=1 . . . n, form the output vector y(k).

Based on the linear model of Equation (16), a multivariable feedbackcontroller (or dynamic compensator) is designed that trades offperformance versus robustness such as thickness, non-uniformity,tracking, noise filtering, pressure saturation limits, etc. Candidatemultivariable linear compensation techniques are Linear QuadraticGaussian (LQG) control, Quantitative Feedback Theory (QFT), andH_(∞)/μ-synthesis. See G. F. Franklin, J. D. Powel, A. Emami-Naeini,Feedback Control of Dynamic Systems, 4^(th) edition, Prentice Hall,2002; D. de Roover, Motion Control of a Wafer Stage—A Design Approachfor Speeding Up IC Production, Ph.D. Dissertation, Mech. Eng. Systemsand Control Group, Delft Univ. of Technology, The Netherlands, 1997; J.H. Vincent, A. Emami-Naeini, N. M. Khraishi, Case Study ComparisonLinear Quadratic Regulator and H _(∞) Control Synthesis, J. Dynamics andControl, Vol. 17, No. 5, pp. 958-965, September-October 1994; Zhou, K.,J. C. Doyle, Essentials of Robust Control, Prentice-Hall, 1998; and J.M. Maciejowski, Multivariable Feedback Design, Addison-Wesley, 1989.

For CMP, compensators can be designed with different techniques. Thedesign technique used is not key to the invention, as long as the finalcompensator does its job of tracking, noise filtering, disturbancecompensation, etc. The final MIMO feedback controller is a critical partof the invention though. To the inventors' best knowledge, this type offeedback control using in-situ sensors is unprecedented anywhere in CMPapplications. The final compensator has a state-space form similar tothat of Equation (16):x _(c)(k+1)=A _(c) x _(c)(k)+B _(c) u _(c)(k),   (17)y _(c)(k)=C _(c) x _(c)(k)+D _(c) u _(c)(k),where the subscript c is indicating the fact that these are allcontroller state variables, inputs, and outputs. Note that thecontroller inputs are typically the CMP model outputs, and thecontroller outputs are the inputs to the CMP model, see FIGS. 2 and 4.

One important advantage of using feedback control based on in-situsensing is the fact that the controller controls the non-uniformityduring the run, i.e. the quality of each single wafer is maximized.Another advantage of feedback control is the fact that the feedbackcontroller can be used to accommodate disturbances such as process noiseand/or drift of machine parameters.

Gain Estimation Model (81)

Based on Equation (12), the gain of the system from pressures to removalrate is determined by the values of relative velocity and Preston'sconstant for the particular process at hand. As mentioned above, theMIMO feedback compensator is designed for linear models in a givenoperating point of the system. If the operating point changessignificantly, the gain of the system may change significantly.Typically, this results in loss of control performance and can lead toinstability of the control loop. To make the controller more robust, aGain Estimation Model 81 (see FIG. 8) was added to the controller thatestimates the gain from the measured thickness data in real-time. If thethickness at each point in time is stored in memory, it isstraightforward to estimate the removal rate of the system at each pointin time by taking the difference of the currently measured thickness andthe previously measured thickness. This gives the left-hand of Equation(12). A nominal linear model was derived based on nominal values for theright-hand of Equation (12). The ratio of these two thus gives anestimate of the gain of the-system. This estimate is used to adjust thecontrol variables in the MIMO Feedback Loop up or down, dependent onwhether the gain is larger than unity or smaller than unity,respectively. This gain estimate is therefore model-based and it doesfeed into the MIMO Feedback Loop, see FIG. 8.

The final output of the Pressure Profile Control Module comes out of aswitch 82, see FIG. 8:

-   -   If the controller is used in an Open-loop (OL) manner, the        measured input pressures coming from a host system are passed on        directly to the output and the controller is not being used.    -   If the controller is used in a Closed-loop (CL) manner, the        pressure output is composed of the offset values, p_(offs)(t),        coming out of the Temperature Control Module (feedforward values        in nominal pressure set point) and the feedback values coming        out of the MIMO Feedback Loop, p_(fb)(t).        Slurry Flow Control Module (44)

The Slurry Flow Control Module 44 controls the slurry flow to thesystem, see FIG. 4. It takes the offset value from the Slurry Flow SlaveLoop 55, which comes out of the Temperature Control Module 41, see FIG.5, and computes the slurry flow set point for the system. Currently, theSlurry Flow Control Module is designed such that slurry flow controlrelieves the pressure control in trying to control process temperature.In other embodiments, the Slurry Flow Control Module may be enhanced byincorporating a slurry transport model 24, as shown in FIG. 2, thatdetermines how slurry flow affects the distributed removal rate. Thisinformation is used to control slurry flow such that wafer profileuniformity is enhanced, in addition to controlling wafer profileuniformity through temperature and pressure control.

Motor Velocity Control Module (43)

The Motor Velocity Control Module 43 controls the motor velocity of boththe wafer spindle (carrier or head) and the platen (or pad), see FIG. 4.It takes the offset value from the Velocity Slave Loop 56, which comesout of the Temperature Control Module, see FIG. 5, and computes thevelocity set points for the DC motors. Currently, the Motor VelocityModule is designed such that motor velocity control relieves thepressure control and slurry flow control in trying to control processtemperature. In other embodiments, the Motor Velocity Control Module isenhanced by incorporating a kinematic model 20, as shown in FIG. 2, thatdetermines how motor velocity affects the distributed removal rate. Thisinformation is used to control wafer and pad motor velocities such thatwafer profile uniformity is enhanced in addition to controlling waferprofile uniformity through temperature and pressure control.

For example, kinematic analysis for a rotational CMP system yields thefollowing expression for relative speed between a point on the wafer andthe pad. See, also, B. U. Yoon, R. P. Young, K. J. In, L. S. Chan, Y. L.Moon, The Effects of Platen and Carrier Rotational Speeds on WithinWafer Non-Uniformity of CMP Removal Rate, 1998 CMP-MIC Conference,IMIC—300P/98/0193, Feb. 19-20, 1998; and D-Z. Chen, B-S. Lee, PatternPlanarization Model of Chemical Mechanical Polishing, J. of theElectrochemical Society, 146(2), 744-748, 1999: $\begin{matrix}{{{v_{rel}\left( {r_{w},t} \right)} = \sqrt{{r_{o}^{2}\omega_{p}^{2}} + {r_{w}^{2}\left( {\omega_{w} - \omega_{p}} \right)}^{2} + {2r_{o}r_{w}{\omega_{p}\left( {\omega_{p} - \omega_{w}} \right)}\cos\quad\omega_{w}t}}},} & (18)\end{matrix}$with r_(o) the distance between the centers of the pad and the wafer,ω_(p) and ω_(W) the rotational speeds of the platen and wafer,respectively, and r_(W) the distance of a point on the wafer from thewafer center. The time-averaged relative speed of a point on the waferat distance r_(W) from the wafer center is now obtained by:$\begin{matrix}{{{{\overset{\_}{v}}_{rel}\left( r_{w} \right)} = {\frac{1}{T_{w}}{\int_{0}^{T_{w}}{{v_{rel}\left( {r_{w},t} \right)}{\mathbb{d}t}}}}},} & (19)\end{matrix}$with T_(w)=2π/ω_(w) the rotational period of the wafer. For givenrotational speeds ω_(p) and ω_(W), and center-to-center distance r_(o),the time-averaged relative velocity can be computed as a function ofwafer radius by solving the integral in Equation (19) numerically. FromEquations (12), (18) and (19) it is seen that the removal rate is afunction of the rotational speeds of pad and wafer, and that uniformremoval rate is obtained if and only if pad and wafer rotational speedsare equal, which is a well known fact in CMP processes. If therotational speeds are not equal to one another, the removal rate isnon-uniform across the wafer in a parabolic shape. The profileuniformity can thus be influenced by independently controlling theplaten and wafer velocity, respectively.

It should be noted that the proposed approach is applicable to CMPsystems with different kinematic motions as well, e.g. a rotational CMPsystem with a sweep arm, or a linear CMP system, or an orbital CMPsystem, etc. In all these cases, Equations (18) and (19) are replaced bytheir corresponding kinematic equations.

Post-Scaling for Temperature Control (45)

The Post-Scaling for Temperature Control Module 45 performs a scaling ofall control variables, i.e. pressures, slurry flow, and motorvelocities, in order to maintain tight temperature control. If any ofthe Modules changes the nominal offset values coming out of theTemperature Control Module, the final process temperature is affected.For example, if the Pressure Profile Control Module decides that itneeds to increase or decrease some of the zone pressure to maintainuniformity, the process temperature is increased or decreased,respectively, and thus deviates from the reference temperature ascomputed in the Temperature Control Module (see FIG. 5). This deviationcan be minimized by post-scaling all zone pressures such that the nettemperature increase or decrease from the individual zone adjustments iscounteracted by total scaling of all pressures. Similarly, otheractuation variables can be used for scaling, such as slurry flow rateand motor velocity, see FIG. 4. In this manner both uniformity andtemperature are controlled. This explains why the scaling blockpreferably comes after the individual Control Modules.

Conclusion

Herein disclosed is an innovative approach for model-based real-timecontrol of CMP systems. The invention provides a method and apparatusthat processes in-situ data from a suite of real-time sensors andproduces real-time commands to multiple actuators, such as appliedpressures, slurry flow, and wafer/pad velocity. A key aspect of theinvention is an integrated model-basedpressure-temperature-velocity-slurry flow control system that includesmany innovations in real-time mode identification, real-time gainestimation, and real-time control. Detailed and accurate componentmodels of the CMP process enable this kind of advanced complexity. Thesolution and methodology are generic and readily allow inclusion of alltypes of sensors.

While the preferred method of control of the invention herein disclosedis in-situ feedback/feedforward control, the invention is not limited tothis type of control only. The disclosed invention can be augmented withother types of control, such as run-to-run control, iterative learningcontrol, adaptive control, etc., without departing from the spirit andscope of the present invention.

It should further be noted that the type of models discussed indisclosed invention are not limited to physics-based mathematicalmodels, but can be replaced by empirical mathematical models, or anycombination of physics-based and empirical mathematical models(semiempirical models), without departing from the spirit and scope ofthe present invention.

Although the invention is described herein with reference to thepreferred embodiment, one skilled in the art will readily appreciatethat other applications may be substituted for those set forth hereinwithout departing from the spirit and scope of the present invention.For example, the invention is readily applicable to CMP machines for usewith wafers of any style, e.g. 150 mm, 200 mm, 300 mm, 400 mm, etc.Further, the invention is applicable to applications of CMP that gobeyond semiconductor fabrication, and includes manufacturing ofread-write heads for hard disks and Micro-Electrical-Mechanical Systems(MEMS) devices. The invention applies to any planarization machine.Secondary applications are traditional precision grinding and polishingwhich include optics/ceramics industries, glass and metal manufacturersas well as in double-sided memory disk media grinding.

Accordingly, the invention should only be limited by the Claims includedbelow.

1. A real-time in-situ model-based chemical-mechanical planarization(CMP) controller, comprising: a dynamic mathematical model of a CMPsystem to be controlled in response to in-situ data from a plurality ofreal-time sensors in said CMP system, said mathematical model comprisingany of a physics-based model and an empirical model; computer simulationmeans for evaluating said mathematical model; and means for validatingsaid mathematical model using any of real-time, in-situ data and ex-situdata from said CMP system.
 2. The controller of claim 1, furthercomprising: a reduced mathematical model for real-time in-situ CMPsystem control.
 3. The controller of claim 1, said mathematical modelcomprising any of: a real-time removal rate model of a CMP system; and areal-time thermal model of a CMP system.
 4. The controller of claim 1,wherein modifications of said CMP system can be evaluated with saidmathematical model via computer simulation prior to any CMP systemmodifications.
 5. A method for producing a real-time in-situ model-basedchemical-mechanical planarization (CMP) controller, comprising the stepsof: generating a dynamic mathematical model of a CMP system to becontrolled in response to in-situ data from a plurality of real-timesensors in said CMP system said mathematical model comprising any of aphysics-based model and an empirical model; evaluating said mathematicalmodel with a computer simulation; and validating said mathematical modelusing any of real-time in-situ data and ex-situ data from said CMPsystem.
 6. The method of claim 5, further comprising the step of:controlling said CMP system in real-time with a reduced mathematicalmodel.
 7. The method of claim 6, said mathematical model comprising anyof: a real-time removal rate model of a CMP system; and a real-timethermal model of a CMP system.
 8. The method of claim 6, furthercomprising the step of: evaluating said CMP system with saidmathematical model via computer simulation prior to any CMP systemmodifications.
 9. In a model-based chemical-mechanical planarization(CMP) controller, said model comprising a dynamic mathematical model ofa CMP system to be controlled in real-time, said mathematical modelcomprising any of a physics-based model, an empirical model, and anycombination thereof, said mathematical model produced by open-loop andclosed-loop techniques comprising use of both a computer simulationmeans for evaluating said mathematical model and means for validatingsaid mathematical model using any of real-time, in-situ data and ex-situdata from said CMP system, said controller comprising: a reducedmathematical model for CMP system control in response to in-situ datafrom a plurality of real-time sensors in said CMP system.